> then you just collect a sufficiently large set of primes, multiply them all together and subtract 1 - bingo.
That doesn't work, but it sounds a bit like the proof that there are an unlimited number of primes.
The proof that there are an unlimited number of primes goes something like this:
Suppose you think there are only a limited number of primes, and you have a list of all the primes. Take all the primes you know about, multiply them all together, then ADD one, to get a new number P. This will be bigger than all the primes you know about. You can't divide P by any of the primes you know about, because you'll always get a remainder of one. So either P is a prime that you didn't know about, or it's composite and it's factors are primes you didn't know about. This proves that your list of primes was incomplete.
This proves that any (finite-length) list of primes will always be incomplete, so there have to be infinitely many primes.
Unfortunately, this process is not a great way of finding new primes, because of the step at the end where you have to figure out if P is prime or not, and factor it if it's not a prime. Factoring big numbers is a very hard problem. (If you can solve that problem, you can break the RSA encryption algorithm).
Biting the hand that feeds IT
